Using the \((O(1),\tilde{O}(k))\) -bicriteria subspace approximation solution found, we design a finer sampling process based on Theorem 45 of [1]} to further pick a subspace of dimension \(\tilde{O}(k^{3}/\epsilon ^2)\)  that contains a \((1+\epsilon )\) -approximate solution for subspace approximation of the matrix \(A(I-BB^\mathsf {T})\) .
